Re: [R] Std. errors of intercept and slope

[Spencer Graves]

See especially any discussion of "the matrix formulation of 
regression".  I'm sure this is in many books.  I'm not familiar with the 
recent literature, but I know it is in Draper and Smith, Applied 
Regression Analysis and Box, Hunter and Hunter, Statistics for 
Experimenters. 

Briefly, suppose we write the regression model as y = X b + e, where y 
and e are N x 1 vectors, X is an N x k matrix, and e is a vector of 
normal, independent errors with standard deviation s.e.  Then the least 
squares and maximum likelihood estimate of b is

     b.hat = (inverse(X' X))*(X'y),

and the covariance matrix for b.hat is

     var(b.hat) = s.e^2 * inverse(X'X). 

I apologize if this is too terse for you;  if so, please see any good 
book on regress. 

hope this helps.  spencer graves

Ben Bolker wrote:

 I'm afraid you're going to have to look it up in a basic statistics 
textbook.

 Ben Bolker

On Fri, 26 Sep 2003, Yao, Minghua wrote:

 

Thanks, Ben.

Could you tell me the formula for calculating this sd., given (x_i, y_i)
(i=1,2,...,N)?
We only have one intercept and slope for them.
-Minghua

-----Original Message-----
From: Ben Bolker [mailto:[EMAIL PROTECTED]
Sent: Friday, September 26, 2003 4:34 PM
To: Yao, Minghua
Cc: R Help (E-mail)
Subject: Re: [R] Std. errors of intercept and slope


 Since the intercept and slope are estimated parameters, they have 
sampling distributions described by their means and standard deviations.  
The s.d. tells you the size of the uncertainty in intercept & in slope.

 This is a pretty basic stats question -- you need to refer to a standard 
textbook or reference material ...

 Ben Bolker

On Fri, 26 Sep 2003, Yao, Minghua wrote:

   

Dear all,

I have the following output generated by linear regression. Since there is
only one regression intercept and one slope for one set of data, what is
     

the
   

meaning of std. error for intercept and that of slope? Thanks in advance.

Sincerely,

Minghua

     

data(thuesen)
attach(thuesen)
lm(short.velocity~blood.glucose)
       

Call:
lm(formula = short.velocity ~ blood.glucose)
Coefficients:
 (Intercept)  blood.glucose  
     1.09781        0.02196  

     

summary(lm(short.velocity~blood.glucose))
       

Call:
lm(formula = short.velocity ~ blood.glucose)
Residuals:
    Min       1Q   Median       3Q      Max 
-0.40141 -0.14760 -0.02202  0.03001  0.43490 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)    1.09781    0.11748   9.345 6.26e-09 ***
blood.glucose  0.02196    0.01045   2.101   0.0479 *  
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

Residual standard error: 0.2167 on 21 degrees of freedom
Multiple R-Squared: 0.1737,     Adjusted R-squared: 0.1343 
F-statistic: 4.414 on 1 and 21 DF,  p-value: 0.0479 

     

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